The equation of the ellipse whose vertices are $( \pm 5,\;0)$ and foci are $( \pm 4,\;0)$ is

The ellipse ${x^2} + 4{y^2} = 4$ is inscribed in a rectangle aligned with the coordinate axes, which in trun is inscribed in another ellipse that passes through the point $(4,0) $  . Then the equation of the ellipse is :

If a tangent having slope of $ - \frac{4}{3}$ to the ellipse $\frac{{{x^2}}}{{18}} + \frac{{{y^2}}}{{32}} = 1$ intersects the major and minor axes in points $A$ and $B$ respectively, then the area of $\Delta OAB$ is equal to .................. $\mathrm{sq. \, units}$ ($O$ is centre of the ellipse)

The acute angle between the pair of tangents drawn to the ellipse $2 x^{2}+3 y^{2}=5$ from the point $(1,3)$ is.

Let $L$ be a tangent line to the parabola $y^{2}=4 x-20$ at $(6,2)$ . If $L$ is also a tangent to the ellipse $\frac{ x ^{2}}{2}+\frac{ y ^{2}}{ b }=1,$ then the value of $b$ is equal to ..... .

If $y = mx + c$ is tangent on the ellipse $\frac{{{x^2}}}{9} + \frac{{{y^2}}}{4} = 1$, then the value of $c$ is